Find the x- and y-intercepts: f(x) = (x^2 - 2x)/(x^2 - 1)

x-intercepts (0, 0), (2, 0); y-intercept (0, 0)

Find the x- and y-intercepts: f(x) = (x^2 + 2x - 8)/(x^2 - 4x - 12)

x-intercepts (-4, 0), (2, 0); y-intercept (0, 2/3)

Find the x- and y-intercepts: f(x) = (3x^2 - 3x - 36)/(x^2 + 3x - 18)

x-intercepts (-3, 0), (4, 0); y-intercept (0, 2)

Find the x- and y-intercepts: f(x) = (3x^2 - 12x + 9)/(x^2 + x - 6)

x-intercepts (1, 0), (3, 0); y-intercept (0, -3/2)

Find the x- and y-intercepts: f(x) = (2x - 8)/(x - 1)

x-intercept (4, 0); y-intercept (0, 8)

Sketch y = (2)/(x - 5) - 2 and give its asymptotes, domain and range.

Asymptotes x = 5 and y = -2;; domain x ≠ 5; range y ≠ -2

Give the domain, intercepts, asymptotes and any hole, then sketch: f(x) = (2(x + 3)(x - 5))/((x + 2)(x - 4))

Domain x ≠ -2, 4; x-intercepts (-3, 0), (5, 0); y-intercept (0, 15/4); vertical asymptotes x = -2, x = 4; horizontal asymptote y = 2; hole none

Find the slant asymptote, the vertical asymptote and any hole: f(x) = (-12x^2 - 27x - 21)/(4x + 5)

y = -3x - 3; x = -5/4; hole none

Worksheets · Calculus

Asymptotes worksheet for Calculus

Calculus restates asymptotes as limits: a vertical asymptote at x = a means f(x) grows without bound as x → a, and a horizontal asymptote y = L means f(x) → L as x → ±∞. Expect holes, intercepts, and arrow notation from one side of each line.

Also taught in Precalculus and Algebra 2, each at its own level and with its own examples.

Select your difficulty

Pick more than one for a sheet that mixes them.

The preview can look cramped on a phone. The PDF and printed sheet come out normal.



Sigma Prep · sigmaprep.io/worksheets Name Asymptotes of Rational Functions Date Period Answer each question. Find the vertical asymptotes Find the vertical asymptotes: $f (x) = \frac{- 2 (x - 6)}{(x - 3) (x + 5)}$ Find the vertical asymptotes: $f (x) = \frac{- 4 (x - 8)}{(x + 7) (x + 8)}$ Find the vertical asymptotes: $f (x) = \frac{- 3 (x - 8)}{(x - 5) (x - 1)}$ Find the vertical asymptotes: $f (x) = \frac{x + 1}{(x - 6) (x - 1)}$ Find the horizontal asymptote Find the horizontal asymptote: $f (x) = \frac{- 9 x^{2} + 4 x + 6}{9 x + 1}$ Find the horizontal asymptote: $f (x) = \frac{9 x^{3} - 8 x}{8 x^{3} - x + 9}$ Find the horizontal asymptote: $f (x) = \frac{- 8 x^{3} - 2 x - 6}{8 x^{3} + 2 x - 2}$ Find the horizontal asymptote: $f (x) = \frac{- 8 x + 9}{7 x^{2}}$ Find the hole Find the hole: $f (x) = \frac{x + 2}{(x + 2) (x + 6) (x + 9)}$ Find the hole: $f (x) = \frac{x + 4}{(x - 8) (x - 4) (x + 4)}$ Find the hole: $f (x) = \frac{x + 8}{(x - 9) (x + 3) (x + 8)}$ Find the hole: $f (x) = \frac{x - 5}{(x - 5) (x - 2) (x + 2)}$ Find the intercepts Find the $x$ - and $y$ -intercepts: $f (x) = \frac{x^{2} - 2 x}{x^{2} - 1}$ Find the $x$ - and $y$ -intercepts: $f (x) = \frac{x^{2} + 2 x - 8}{x^{2} - 4 x - 12}$ Find the $x$ - and $y$ -intercepts: $f (x) = \frac{3 x^{2} - 3 x - 36}{x^{2} + 3 x - 18}$ Find the $x$ - and $y$ -intercepts: $f (x) = \frac{3 x^{2} - 12 x + 9}{x^{2} + x - 6}$ Describe behavior near an asymptote $f (x) = \frac{- 5}{{(x - 5)}^{2}}$ . As $x \to 5^{-}$ , $f (x) \to$ ? $f (x) = \frac{- 2 x + 4}{x - 1}$ . As $x \to \infty$ , $f (x) \to$ ? $f (x) = \frac{x - 1}{x + 6}$ . As $x \to \infty$ , $f (x) \to$ ? $f (x) = \frac{4 x - 1}{x - 1}$ . As $x \to - \infty$ , $f (x) \to$ ? Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

Sigma Prep · sigmaprep.io/worksheets

Name

Asymptotes of Rational Functions

Date Period

Answer each question.

Find the vertical asymptotes
Find the vertical asymptotes:
$f (x) = \frac{- 2 (x - 6)}{(x - 3) (x + 5)}$
Find the vertical asymptotes:
$f (x) = \frac{- 4 (x - 8)}{(x + 7) (x + 8)}$
Find the vertical asymptotes:
$f (x) = \frac{- 3 (x - 8)}{(x - 5) (x - 1)}$
Find the vertical asymptotes:
$f (x) = \frac{x + 1}{(x - 6) (x - 1)}$

Find the horizontal asymptote
Find the horizontal asymptote:
$f (x) = \frac{- 9 x^{2} + 4 x + 6}{9 x + 1}$
Find the horizontal asymptote:
$f (x) = \frac{9 x^{3} - 8 x}{8 x^{3} - x + 9}$
Find the horizontal asymptote:
$f (x) = \frac{- 8 x^{3} - 2 x - 6}{8 x^{3} + 2 x - 2}$
Find the horizontal asymptote:
$f (x) = \frac{- 8 x + 9}{7 x^{2}}$

Find the hole
Find the hole:
$f (x) = \frac{x + 2}{(x + 2) (x + 6) (x + 9)}$
Find the hole:
$f (x) = \frac{x + 4}{(x - 8) (x - 4) (x + 4)}$
Find the hole:
$f (x) = \frac{x + 8}{(x - 9) (x + 3) (x + 8)}$
Find the hole:
$f (x) = \frac{x - 5}{(x - 5) (x - 2) (x + 2)}$

Find the intercepts
Find the $x$ - and $y$ -intercepts:
$f (x) = \frac{x^{2} - 2 x}{x^{2} - 1}$
Find the $x$ - and $y$ -intercepts:
$f (x) = \frac{x^{2} + 2 x - 8}{x^{2} - 4 x - 12}$
Find the $x$ - and $y$ -intercepts:
$f (x) = \frac{3 x^{2} - 3 x - 36}{x^{2} + 3 x - 18}$
Find the $x$ - and $y$ -intercepts:
$f (x) = \frac{3 x^{2} - 12 x + 9}{x^{2} + x - 6}$

Describe behavior near an asymptote
$f (x) = \frac{- 5}{{(x - 5)}^{2}}$ . As $x \to 5^{-}$ , $f (x) \to$ ?
$f (x) = \frac{- 2 x + 4}{x - 1}$ . As $x \to \infty$ , $f (x) \to$ ?
$f (x) = \frac{x - 1}{x + 6}$ . As $x \to \infty$ , $f (x) \to$ ?
$f (x) = \frac{4 x - 1}{x - 1}$ . As $x \to - \infty$ , $f (x) \to$ ?



Sigma Prep · sigmaprep.io/worksheets Name Asymptotes of Rational FunctionsAnswer keyVersion 2 Date Period Answer each question. Find the vertical asymptotes Find the vertical asymptotes: $f (x) = \frac{- 2 (x - 6)}{(x - 3) (x + 5)}$ $x = - 5, x = 3$ Find the vertical asymptotes: $f (x) = \frac{- 4 (x - 8)}{(x + 7) (x + 8)}$ $x = - 8, x = - 7$ Find the vertical asymptotes: $f (x) = \frac{- 3 (x - 8)}{(x - 5) (x - 1)}$ $x = 1, x = 5$ Find the vertical asymptotes: $f (x) = \frac{x + 1}{(x - 6) (x - 1)}$ $x = 1, x = 6$ Find the horizontal asymptote Find the horizontal asymptote: $f (x) = \frac{- 9 x^{2} + 4 x + 6}{9 x + 1}$ None Find the horizontal asymptote: $f (x) = \frac{9 x^{3} - 8 x}{8 x^{3} - x + 9}$ $y = \frac{9}{8}$ Find the horizontal asymptote: $f (x) = \frac{- 8 x^{3} - 2 x - 6}{8 x^{3} + 2 x - 2}$ $y = - 1$ Find the horizontal asymptote: $f (x) = \frac{- 8 x + 9}{7 x^{2}}$ $y = 0$ Find the hole Find the hole: $f (x) = \frac{x + 2}{(x + 2) (x + 6) (x + 9)}$ $(- 2, \frac{1}{28})$ Find the hole: $f (x) = \frac{x + 4}{(x - 8) (x - 4) (x + 4)}$ $(- 4, \frac{1}{96})$ Find the hole: $f (x) = \frac{x + 8}{(x - 9) (x + 3) (x + 8)}$ $(- 8, \frac{1}{85})$ Find the hole: $f (x) = \frac{x - 5}{(x - 5) (x - 2) (x + 2)}$ $(5, \frac{1}{21})$ Find the intercepts Find the $x$ - and $y$ -intercepts: $f (x) = \frac{x^{2} - 2 x}{x^{2} - 1}$ $x$ -intercepts $(0, 0)$ , $(2, 0)$ ; $y$ -intercept $(0, 0)$ Find the $x$ - and $y$ -intercepts: $f (x) = \frac{x^{2} + 2 x - 8}{x^{2} - 4 x - 12}$ $x$ -intercepts $(- 4, 0)$ , $(2, 0)$ ; $y$ -intercept $(0, \frac{2}{3})$ Find the $x$ - and $y$ -intercepts: $f (x) = \frac{3 x^{2} - 3 x - 36}{x^{2} + 3 x - 18}$ $x$ -intercepts $(- 3, 0)$ , $(4, 0)$ ; $y$ -intercept $(0, 2)$ Find the $x$ - and $y$ -intercepts: $f (x) = \frac{3 x^{2} - 12 x + 9}{x^{2} + x - 6}$ $x$ -intercepts $(1, 0)$ , $(3, 0)$ ; $y$ -intercept $(0, - \frac{3}{2})$ Describe behavior near an asymptote $f (x) = \frac{- 5}{{(x - 5)}^{2}}$ . As $x \to 5^{-}$ , $f (x) \to$ ? $- \infty$ $f (x) = \frac{- 2 x + 4}{x - 1}$ . As $x \to \infty$ , $f (x) \to$ ? $- 2$ $f (x) = \frac{x - 1}{x + 6}$ . As $x \to \infty$ , $f (x) \to$ ? $1$ $f (x) = \frac{4 x - 1}{x - 1}$ . As $x \to - \infty$ , $f (x) \to$ ? $4$ Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

Sigma Prep · sigmaprep.io/worksheets

Name

Asymptotes of Rational FunctionsAnswer keyVersion 2

Date Period

Answer each question.

Find the vertical asymptotes
Find the vertical asymptotes:
$f (x) = \frac{- 2 (x - 6)}{(x - 3) (x + 5)}$ $x = - 5, x = 3$
Find the vertical asymptotes:
$f (x) = \frac{- 4 (x - 8)}{(x + 7) (x + 8)}$ $x = - 8, x = - 7$
Find the vertical asymptotes:
$f (x) = \frac{- 3 (x - 8)}{(x - 5) (x - 1)}$ $x = 1, x = 5$
Find the vertical asymptotes:
$f (x) = \frac{x + 1}{(x - 6) (x - 1)}$ $x = 1, x = 6$

Find the horizontal asymptote
Find the horizontal asymptote:
$f (x) = \frac{- 9 x^{2} + 4 x + 6}{9 x + 1}$ None
Find the horizontal asymptote:
$f (x) = \frac{9 x^{3} - 8 x}{8 x^{3} - x + 9}$ $y = \frac{9}{8}$
Find the horizontal asymptote:
$f (x) = \frac{- 8 x^{3} - 2 x - 6}{8 x^{3} + 2 x - 2}$ $y = - 1$
Find the horizontal asymptote:
$f (x) = \frac{- 8 x + 9}{7 x^{2}}$ $y = 0$

Find the hole
Find the hole:
$f (x) = \frac{x + 2}{(x + 2) (x + 6) (x + 9)}$ $(- 2, \frac{1}{28})$
Find the hole:
$f (x) = \frac{x + 4}{(x - 8) (x - 4) (x + 4)}$ $(- 4, \frac{1}{96})$
Find the hole:
$f (x) = \frac{x + 8}{(x - 9) (x + 3) (x + 8)}$ $(- 8, \frac{1}{85})$
Find the hole:
$f (x) = \frac{x - 5}{(x - 5) (x - 2) (x + 2)}$ $(5, \frac{1}{21})$

Find the intercepts
Find the $x$ - and $y$ -intercepts:
$f (x) = \frac{x^{2} - 2 x}{x^{2} - 1}$ $x$ -intercepts $(0, 0)$ , $(2, 0)$ ; $y$ -intercept $(0, 0)$
Find the $x$ - and $y$ -intercepts:
$f (x) = \frac{x^{2} + 2 x - 8}{x^{2} - 4 x - 12}$ $x$ -intercepts $(- 4, 0)$ , $(2, 0)$ ; $y$ -intercept $(0, \frac{2}{3})$
Find the $x$ - and $y$ -intercepts:
$f (x) = \frac{3 x^{2} - 3 x - 36}{x^{2} + 3 x - 18}$ $x$ -intercepts $(- 3, 0)$ , $(4, 0)$ ; $y$ -intercept $(0, 2)$
Find the $x$ - and $y$ -intercepts:
$f (x) = \frac{3 x^{2} - 12 x + 9}{x^{2} + x - 6}$ $x$ -intercepts $(1, 0)$ , $(3, 0)$ ; $y$ -intercept $(0, - \frac{3}{2})$

Describe behavior near an asymptote
$f (x) = \frac{- 5}{{(x - 5)}^{2}}$ . As $x \to 5^{-}$ , $f (x) \to$ ? $- \infty$
$f (x) = \frac{- 2 x + 4}{x - 1}$ . As $x \to \infty$ , $f (x) \to$ ? $- 2$
$f (x) = \frac{x - 1}{x + 6}$ . As $x \to \infty$ , $f (x) \to$ ? $1$
$f (x) = \frac{4 x - 1}{x - 1}$ . As $x \to - \infty$ , $f (x) \to$ ? $4$

How to do these

$f (x) = \frac{3}{{(x - 2)}^{2}}$ . As $x \to 2^{-}$ , $f (x) \to$ ?

The bottom goes to 0 as x approaches 2.
It is squared, so it stays positive from either side.
3 over a tiny positive number is large and positive.

The answer is $\infty$ .

Where students go wrong

Taking the sign from the side alone. A squared factor in the bottom is positive from both sides, so the sign comes from the top: 3/(x - 2)² goes to ∞ from the left and from the right.

Also called vertical asymptotes, horizontal asymptotes, slant asymptote, holes in a graph or graphs of rational functions.

What you can put on this worksheet

Vertical asymptotes: Find the vertical asymptotes:
$f (x) = \frac{x - 1}{x - 4}$ → $x = 4$
The horizontal asymptote: Find the horizontal asymptote:
$f (x) = \frac{- 3 x^{2} + 3}{2 x^{2} + x}$ → $y = - \frac{3}{2}$
Holes: Find the hole:
$f (x) = \frac{x - 3}{(x - 3) (x - 1)}$ → $(3, \frac{1}{2})$
Slant asymptotes: Find the slant asymptote:
$f (x) = \frac{2 x^{2} + 10 x + 14}{x + 3}$ → $y = 2 x + 4$
Intercepts: Find the $x$ - and $y$ -intercepts:
$f (x) = \frac{2 x - 8}{x - 1}$ → $x$ -intercept $(4, 0)$ ; $y$ -intercept $(0, 8)$
Arrow notation near an asymptote: $f (x) = \frac{1}{x - 4}$ . As $x \to 4^{+}$ , $f (x) \to$ ? → $\infty$
Sketch a/(x - h) + k, with domain and range: Sketch $y = \frac{2}{x - 5} - 2$ and give its asymptotes, domain and range. → Asymptotes $x = 5$ and $y = - 2$ ;
domain $x \neq 5$ ; range $y \neq - 2$
Quadratic over quadratic: every feature, then sketch: Give the domain, intercepts, asymptotes and any hole, then sketch:
$f (x) = \frac{2 (x + 3) (x - 5)}{(x + 2) (x - 4)}$ → Domain $x \neq - 2, 4$ ; $x$ -intercepts $(- 3, 0), (5, 0)$ ; $y$ -intercept $(0, \frac{15}{4})$ ; vertical asymptotes $x = - 2, x = 4$ ; horizontal asymptote $y = 2$ ; hole none
Slant asymptotes over ax + b, with a hole: Find the slant asymptote, the vertical asymptote and any hole:
$f (x) = \frac{- 12 x^{2} - 27 x - 21}{4 x + 5}$ → $y = - 3 x - 3$ ; $x = - \frac{5}{4}$ ; hole none

Questions about these worksheets

Yes. Every asymptotes sheet is free to print and free to download as a PDF. There is no account to make, no email to hand over and no limit on how many you take.

Yes. The answer key prints on a second page, and you can choose whether it shows each question beside its answer or just the answers in a list.

Yes. Print straight from the page, or download the PDF and print that. The sheet is laid out for paper rather than squeezed off a screen, so there is room to work under each question, and the answer key comes out on its own page.

Calculus is usually taken around 12th grade or a first college course, though schools vary and plenty of students meet it earlier or later. Pick the difficulty rather than the grade: the easy level suits a first lesson on it, the hard level suits review before a test.

Yes, as many as you like. The questions are built fresh each time rather than picked from a fixed set of files, so pressing Generate gives a new sheet. That is what you want for a second class, a retake, or two students sitting next to each other.

That is the closest comparison, and Kuta Software is good software. It is also paid software you install on a computer. These worksheets run in a browser tab for free, with no account and nothing to install. Like Kuta, the asymptotes questions are generated when you ask for them rather than pulled from a fixed set of files, so two classes never get the same sheet, and the answer key comes with it.

Yes. There are three levels, and you can tick more than one for a sheet that mixes them, in which case the questions come out easiest first. The harder levels are not just larger numbers: they ask for something the easy ones do not.

You pick from 9: vertical asymptotes, the horizontal asymptote, holes, slant asymptotes, intercepts, arrow notation near an asymptote, sketch a/(x - h) + k, with domain and range, quadratic over quadratic: every feature, then sketch and slant asymptotes over ax + b, with a hole. Tick as many as you want and set how many of each, or let it spread them evenly.

Calling a canceled factor an asymptote. If it divides out of the top as well, the graph has a hole there, not a wall, and the sheet is testing whether you noticed.

Asymptotes worksheet for Calculus

Select your difficulty

Find the vertical asymptotes

Find the horizontal asymptote

Find the hole

Find the intercepts

Describe behavior near an asymptote

Find the vertical asymptotes

Find the horizontal asymptote

Find the hole

Find the intercepts

Describe behavior near an asymptote

How to do these

Where students go wrong

What you can put on this worksheet

Questions about these worksheets

Are these asymptotes worksheets free?+

Do the asymptotes worksheets come with an answer key?+

Are these asymptotes worksheets printable?+

What grade are these asymptotes worksheets for?+

Can I get a different version of the same asymptotes worksheet?+

Is this a free alternative to Kuta Software for asymptotes?+

Can I choose how hard the asymptotes questions are?+

What kinds of question are on a asymptotes worksheet?+

What do students get wrong with asymptotes?+